By Richard O. Moore
The stated goal of SIAM Activity Groups (SIAGs) is to “provide a more focused forum for SIAM members interested in exploring one of the areas of applied mathematics, computational science, or applications.” SIAM’s new blog space seems the perfect venue for some casual exploration of these areas, starting with crude sketches of their domains. As the Secretary for the SIAG on Nonlinear Waves and Coherent Structures, I’ll try to wrap my arms around this fraught task.
Many of us still regard John Scott Russell’s celebrated ride along the Union Canal as the birthing event of the field of nonlinear waves. Solitons (bright and dark), breathers, fronts, kinks, vortices, instantons—these localized phenomena continue to fascinate due to their sheer improbability in spatially extended systems, and due to the convenience they offer in quantifying behavior in an ostensibly infinite-dimensional model through a finite-dimensional dynamical system.
The first questions asked still focus on existence and stability, but from there the direction of investigation is often motivated by application. Can we promote, control, arrest or mitigate against the action of a nonlinear wave? Can we encode information on it? Can we predict with certainty when it will form and what impact it will have? Can we explain why and how a pattern emerges from an otherwise uniform state?
Patterns and localized phenomena also arise as unexpected coherence in an otherwise incoherent or random system. This is the case in fluid and plasma turbulence where long-lived coherent structures, typically vortices, develop out of a turbulent background and survive despite being embedded within it. Indeed the term “coherent structure” first appeared in this context. This side of our activity group has seen much growth over the last decade or two due to the discovery of the importance of coherent structures in the transition to turbulence and the role they play in dynamical recurrence in turbulent shear flows.
At the same time, advances in computation have permitted the first direct numerical simulations of fluid flows at unprecedented Reynolds numbers, while large-scale Monte Carlo simulations have led to new advances in our understanding of the effects of noise on nonlinear systems with many degrees of freedom. In many model equations describing systems of this type, new bifurcations result from the homogenized effect of a random term, while in other systems the disorder dominates the leading-order behavior to produce such phenomena as Anderson localization and time-reversed refocusing. Moving away from equilibrium statistics one also sees interesting behavior at the probabilistic margins, with instantons playing an arguably large role in the occurrence of freak waves in the ocean or bit errors in optical communications.
Being a group defined by phenomena rather than a mathematical category, it is no surprise that our members use a wide variety of techniques. Asymptotic and perturbative methods have always been central to the activity group, as have dynamical systems, bifurcation theory and integrable systems. As in many areas, the tool whose usefulness has grown the fastest is scientific computing, providing the ability to simulate phenomena on incredibly fine spatiotemporal scales. This growth in computing power, including hardware advances such as massive parallelization and innovation in numerical methods, has allowed direct comparisons between particle simulations and mean-field approximations as well as statistical resolution of rare events in stochastic systems. Relative to the steady growth in computing power, new analytical tools come online in fits and starts, but advances in such arenas as asymptotics beyond all orders, geometric methods and symmetry analysis have revolutionized the field.
Polar vortex on January 6, 2014, in terms of 500 millibar height (false color) and sea level pressure (black contours), superimposed on white outline of North America in lower middle of figure. Source: Pacific Marine Environmental Laboratory, National Oceanic and Atmospheric Administration (http://www.pmel.noaa.gov/arctic.shtml).
Looking forward, one might anticipate that new growth emerges primarily from key application areas. New materials with complex internal structure allow for refined control of acoustic and optical waves in solids, with applications ranging from drug delivery to sensor technology to data storage. Nonlinear waves in biology and ecology abound, from morphogenic patterns to swarms of insects to heart and brain waves, and this area will continue to grow as our understanding of living organisms and systems improves. Climate and weather models, particularly their atmospheric and oceanographic components, rely heavily on an understanding of nonlinear waves and emergent phenomena, and this area will grow in importance as we look for ways to mitigate the alarming impact of carbon in the air and oceans.
In short, it is a very exciting time to be studying nonlinear waves and coherent structures, and the future seems very bright for our activity group. In the immediate future, of course, is the SIAM Conference on Nonlinear Waves and Coherent Structures from August 11 to 14, 2014, at Cambridge University in the UK. We look forward to seeing many of you at NW14 and in the audience for this years’s Martin Kruskal Lecture!
Acknowledgments: This post was prepared with help from SIAG/NWCS officers Edgar Knobloch, Margaret Beck, and Paul Milewski.
Richard O. Moore is an associate professor of mathematical sciences at the New Jersey Institute of Technology.